Theorem univ 4567

Description: The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
univ	⊢ ∪ V = V

Proof of Theorem univ

Step	Hyp	Ref	Expression
1		pwv 3887	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 3898	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 4303	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2252	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1395  Vcvv 2799  𝒫 cpw 3649  ∪ cuni 3888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-uni 3889

This theorem is referenced by: (None)